Enjoy these links before you read the main body of the text. Also think about visiting my Jazz Videos page above for a regularly-updated selection of good jazz videos. Apologies for links only; YT doesn't seem to find the video I want when I try to add a video. The joys of modern technology.
This is a question that you must not answer too quickly. It is a question that you must think about because the answer is different for different people. Some people play jazz piano for personal enjoyment, others play it just to perform publically and make money from it, others use it to develop their piano skills and others play it for the pleasure of not having to read music. All are very valid reasons.
For me, playing jazz piano is a freedom of sound. I love that I can play a particular jazz number in a variety of ways, choosing to highlight particular chord types or melodic ideas which work in one style, but not in another. It is the delight in knowing that I see the piano in a unique way and that I am not bound to strict rules.
Jazz piano represents a history of humanity which, whilst being negative in its origins (slave trade), enabled particular individuals to play their instrument in a way which had never been played before. Take Louis Armstrong for example; he revolutionised how to play the trumpet. Art Tatum revolutionised improvisation to a level that most will never achieve (even OP placed himself second to AT, which I disagree with - OP was much more musical and varied than AT, but I digress). In a way, the freedom of jazz can be compared with the freedom of people throughout the 20th centurty; being unbound by crazy black/white laws in the same was as no longer being bound by notes on a page from which no player should depart.
Jazz piano provides the opportunity to play how you feel on a given day; playing classical piano does not provide this. A piece of such music is the same no matter when it is played; jazz let's you be you, and for me, that is very important. Is it moral to improvise the classical repertiore?
The next time you go to your piano to play jazz, before you start playing, just think about the chance that this music is giving you. You don't need to pull out any music, you don't need to even remember a particular song to play; you can simply acknowledge how you feel and improvise accordingly.
Do you feel a little lonely and miserable today because of something that has happened recently or because the weather is a bit miserable outside? Go for some minor 9ths or sus4 chords into M7s a 4th up. Then, try to brighten up your feelings by creating some smooth-sounding II-V-Im(new II)-V-Im(new II)-V, etc.. this progression where the final chord of the II-V-I is surprisingly a new IIm in a new key is quite interesting sounding. Get some nice improvisations over that sequence in some new keys and feel the freedom. Feel happier that you are able to even do this and understand what is going on. It should cheer you up a bit.
Is your day going well? Get out some 13ths and a nice blue scale run. Try different keys' blues scales so that you can come to hear the differences each blues scale key/chord sounds; you will be quite delighted (try B flat blues scale of F13, for example, then try D blues against F! Experiment). Move from M7 to d7 chords, play a few happy-sounding 6ths, etc. Improvise with lots of b5 in there, with some minor 3rd to major 3rd slides (grace notes). Enjoy this freedom.
I hope I have helepd to change your day a little bit. Do not forget about my eBook: A Philosophical Approach to Jazz Piano, in which you can read away from the piano and develop your inside awareness and appreciation for jazz piano so that you can become a more proficient Purposeful Pianist.